The proof is similar to the proof of [1]}.
We can choose a rational polyhedral cone \(\sigma \) , a map of cones \(u: \mathbb {R}_{\ge 0}\rightarrow \sigma \)  and a stable tropical curve \(\widetilde{\Gamma }/\sigma \)  over \(\sigma \)  such that \((\widetilde{\Gamma })=(\overline{\Gamma })\)  and the metric \(d_{\overline{\Gamma }}\)  of \(\overline{\Gamma }\)  is equal to the
following composition
\(d_{\overline{\Gamma }}\colon E((\overline{\Gamma }))=E((\widetilde{\Gamma })) {d_{\widetilde{\Gamma }}} S_{\sigma }\longrightarrow \sigma ^{\vee }{u^{\vee }} (\mathbb {R}_{\ge 0})^{\vee }\cong \mathbb {R}_{\ge 0}.\) 
This is equivalent to saying that the morphism of real cone stacks \(\mathbb {R}_{\ge 0}\xrightarrow{} \mathcal {M}^{trop, \mathbb {R}}_{g,n}\)  corresponding to the tropical curve \(\overline{\Gamma }\in \big |\mathcal {M}^{trop, \mathbb {R}}_{g,n}\big |(\star )=\mathcal {M}^{trop, \mathbb {R}}_{g,n}(\mathbb {R}_{\ge 0})\)  factors as
\(\mathbb {R}_{\ge 0}{u} \sigma {\widetilde{\Gamma }} \mathcal {M}^{trop, \mathbb {R}}_{g,n}\) 
where the last morphism \(\sigma \xrightarrow{} \mathcal {M}^{trop, \mathbb {R}}_{g,n}\)  corresponds to the tropical curve \(\widetilde{\Gamma }\)  seen as an element of \(\mathcal {M}^{trop, \mathbb {R}}_{g,n}(\sigma )\) .
Since the fiber of \(\Phi ^{\operatorname{trop}}\)  over \(\sigma \xrightarrow{} \mathcal {M}^{trop}_{g,n}\)  is equal to \(\operatorname{Jac}_{\widetilde{\Gamma }/\sigma }\rightarrow \sigma \)  by Theorem REF , we get that the fiber of \(\Phi ^{\operatorname{trop},\mathbb {R}}\)  over \(\mathbb {R}_{\ge 0}\xrightarrow{} \mathcal {M}^{trop, \mathbb {R}}_{g,n}\)  is equal to
\(\mathbb {R}_{\ge 0}\times _{\sigma } \left(\sigma \times _{\mathcal {M}^{trop, \mathbb {R}}_{g,n}} \mathcal {J}^{trop, \mathbb {R}}_{g,n}\right)=\mathbb {R}_{\ge 0}\times _{\sigma } (\operatorname{Jac}_{\widetilde{\Gamma }/\sigma })^{\mathbb {R}}\longrightarrow \mathbb {R}_{\ge 0},\) 
where \((\operatorname{Jac}_{\widetilde{\Gamma }/\sigma })^{\mathbb {R}}\)  is the real cone space associated to \(\operatorname{Jac}_{\widetilde{\Gamma }/\sigma }\) .
Passing to the associated topological stacks, we get that the desired fiber \(|\Phi ^{\operatorname{trop}}|^{-1}(\overline{\Gamma })\)  is equal to the fiber over \(1\in |\mathbb {R}_{\ge 0}|\xrightarrow{} |\sigma |\)  of the morphism of topological spaces \(|\operatorname{Jac}_{\widetilde{\Gamma }/\sigma }|\rightarrow |\sigma |\) , which is equal to
\(\operatorname{Jac}_{\overline{\Gamma }}\)  as it follows by comparing Definitions REF  and REF  and the fact that \(u^*(\widetilde{\Gamma })=\overline{\Gamma }\)  by construction.
